Is there any way to get variable importance with Keras?

*Edited to include relevant code to implement permutation importance.

I answered a similar question at Feature Importance Chart in neural network using Keras in Python. It does implement what Teque5 mentioned above, namely shuffling the variable among your sample or permutation importance using the ELI5 package.

from keras.wrappers.scikit_learn import KerasClassifier, KerasRegressor
import eli5
from eli5.sklearn import PermutationImportance

def base_model():
    model = Sequential()        
    ...
    return model

X = ...
y = ...

my_model = KerasRegressor(build_fn=basemodel, **sk_params)    
my_model.fit(X,y)

perm = PermutationImportance(my_model, random_state=1).fit(X,y)
eli5.show_weights(perm, feature_names = X.columns.tolist())

Since everything will be mixed up along the network, the first layer alone can't tell you about the importance of each variable. The following layers can also increase or decrease their importance, and even make one variable affect the importance of another variable. Every single neuron in the first layer itself will give each variable a different importance too, so it's not something that straightforward.

I suggest you do model.predict(inputs) using inputs containing arrays of zeros, making only the variable you want to study be 1 in the input.

That way, you see the result for each variable alone. Even though, this will still not help you with the cases where one variable increases the importance of another variable.